Normalizing Flows (NFs) are powerful and efficient models for density estimation. When modeling densities on manifolds, NFs can be generalized to injective flows but the Jacobian determinant becomes computationally prohibitive. Current approaches either consider bounds on the log-likelihood or rely on some approximations of the Jacobian determinant. In contrast, we propose injective flows for star-like manifolds and show that for such manifolds we can compute the Jacobian determinant exactly and efficiently, with the same cost as NFs. This aspect is particularly relevant for variational inference settings, where no samples are available and only some unnormalized target is known. Among many, we showcase the relevance of modeling densities on star-like manifolds in two settings. Firstly, we introduce a novel Objective Bayesian approach for penalized likelihood models by interpreting level-sets of the penalty as star-like manifolds. Secondly, we consider probabilistic mixing models and introduce a general method for variational inference by defining the posterior of mixture weights on the probability simplex.

翻译：归一化流（NFs）是用于密度估计的强大且高效的模型。在对流形上的密度进行建模时，NFs可以推广为单射流，但此时雅可比行列式的计算变得计算上难以处理。当前的方法要么考虑对数似然的边界，要么依赖于对雅可比行列式的某些近似。相比之下，我们提出了针对星形流形的单射流，并证明对于此类流形，我们可以精确且高效地计算雅可比行列式，其计算成本与NFs相同。这一特性在变分推断场景中尤为重要，因为此类场景下没有样本可用，仅已知某个未归一化的目标。我们通过多种案例展示了在星形流形上建模密度的相关性，并重点介绍了两种场景。首先，我们通过将惩罚项的水平集解释为星形流形，为惩罚似然模型引入了一种新颖的客观贝叶斯方法。其次，我们考虑概率混合模型，并通过在概率单纯形上定义混合权重的后验分布，引入了一种通用的变分推断方法。